Reincarnation is about the fact or theory that each of us experiences not just one life, but a very long series of lives, one after another, perhaps with breaks in between in what is called in the East the 'Bardo.'
It is a very old idea,  present in Eastern religions and, in the West, going back at least to Plato, in about 380B.C. This great age actually creates problems; if you want to explore reincarnation through scholarship you have to understand either the ancient Greek or the ancient Indian mind, or both.

If you want a very simple summary, I recommend Wikipedia. There is also quite a nice page, with rather good pictures, on PAST LIFE REGRESSIONS & WORKSHOPS IN GLASTONBURY. Obviously this is someone else's commercial site, I have no problem drawing it to your attention, although I have no connection with it other than approving the historical notes, particularly on Arnall Bloxham.
The modern knowledge of reincarnation largely comes from 20th century hypnotherapists, of which I will mention 3: Michael Newton (biography, Institute) Arthur Guirdham (Answers.com) and Brian Weiss (on YouTube). Coming from medical backgrounds, such people found that treatment led to regressing their clients into past lives, to find the source of their current problems. So, a great deal of what we know today is, in fact, from the treatment of people who may have been adversely affected, in their present lives, by what they brought forward from past lives.
Another area of insight is of children who insist they have memories of being someone else. When parents, driven to desperation, investigate they find the facts of their child's stories to be true. Of the several famous cases of this sort, at least one is on YouTube. One of the best known investigators of such stories was Dr. Ian Stevenson (assessment of his evidence). Of course, the very best known story of child past-life memory concerns how the Tibetan Dalai Lama is selected - the child who can show true memory of being the last leader of Tibetan Buddhism is made its next leader.
There have always been remarkable individuals who claimed to remember past lives; one of these is George Smith Patton, Jr. (November 11,1885 – December 21,1945) a United States Army officer best known for his leadership as a general during World War II, in particular his victory in the Battle of the Bulge in 1944; he ascribed this to his memory of serving as a marshal of France in the retreat from Moscow in 1812. If anyone should ask what is the value of remembering past lives, the answer, for several thousand American and British soldiers, might be they survived, but would have died without Patton's memory. Here is Patton's poem, Throuh a Glass Darkly
My wife takes the view there are good reasons why we don't remember past lives. Clearly there are reasons, I take the view that if you can remember them the benefit can be extraordinary, witness the thousands of patients who pay to go into past-life regression therapy.

Some famous believers in Reincarnation   Here are some famous quotes

Dare you explore further?
The sub-pages under Reincarnation are linked here, they have further links to Franz Kafka and YouTube, and they deal with:
First steps, Inspiration and Imagination,
The reasons we don't remember,
Discovering your own past lives.
Remember: these pages are far from definitive, there will always be much more to know, but, with your help, we shall uncover more as we go.
Exploring further is the business of the Internet radio show, Reincarnation, the Secrets of Past Lives.
It is also part of the business of Metaphysics, which I have put under the heading Reality. As you start to explore, more questions arise, and I try to put these in a context which makes sense to me. You will find there are many schemes and schools which have done the same thing, and still do. Before getting to that it may be easiest to look at Spirit.